[毕业设计精品论文]数形结合思想在解题中的应用 外文

DEPARTMENTOFMATHEMATICSEDUCATIONJWILSON,EMT669THEPYTHAGOREANTHEOREMBYSTEPHANIEJMORRISTHEPYTHAGOREANTHEOREMWASONEOFTHEEARLIESTTHEOREMSKNOWNTOANCIENTCIVILIZATIONSTHISFAMOUSTHEOREMISNAMEDFORTHEGREEKMATHEMATICIANANDPHILOSOPHER,PYTHAGORASPYTHAGORASFOUNDEDTHEPYTHAGOREANSCHOOLOFMATHEMATICSINCORTONA,AGREEKSEAPORTINSOUTHERNITALYHEISCREDITEDWITHMANYCONTRIBUTIONSTOMATHEMATICSALTHOUGHSOMEOFTHEMMAYHAVEACTUALLYBEENTHEWORKOFHISSTUDENTSTHEPYTHAGOREANTHEOREMISPYTHAGORASMOSTFAMOUSMATHEMATICALCONTRIBUTIONACCORDINGTOLEGEND,PYTHAGORASWASSOHAPPYWHENHEDISCOVEREDTHETHEOREMTHATHEOFFEREDASACRIFICEOFOXENTHELATERDISCOVERYTHATTHESQUAREROOTOF2ISIRRATIONALANDTHEREFORE,CANNOTBEEXPRESSEDASARATIOOFTWOINTEGERS,GREATLYTROUBLEDPYTHAGORASANDHISFOLLOWERSTHEYWEREDEVOUTINTHEIRBELIEFTHATANYTWOLENGTHSWEREINTEGRALMULTIPLESOFSOMEUNITLENGTHMANYATTEMPTSWEREMADETOSUPPRESSTHEKNOWLEDGETHATTHESQUAREROOTOF2ISIRRATIONALITISEVENSAIDTHATTHEMANWHODIVULGEDTHESECRETWASDROWNEDATSEATHEPYTHAGOREANTHEOREMISASTATEMENTABOUTTRIANGLESCONTAININGARIGHTANGLETHEPYTHAGOREANTHEOREMSTATESTHAT“THEAREAOFTHESQUAREBUILTUPONTHEHYPOTENUSEOFARIGHTTRIANGLEISEQUALTOTHESUMOFTHEAREASOFTHESQUARESUPONTHEREMAININGSIDES“FIGURE1ACCORDINGTOTHEPYTHAGOREANTHEOREM,THESUMOFTHEAREASOFTHETWOREDSQUARES,SQUARESAANDB,ISEQUALTOTHEAREAOFTHEBLUESQUARE,SQUARECTHUS,THEPYTHAGOREANTHEOREMSTATEDALGEBRAICALLYISFORARIGHTTRIANGLEWITHSIDESOFLENGTHSA,B,ANDC,WHERECISTHELENGTHOFTHEHYPOTENUSEALTHOUGHPYTHAGORASISCREDITEDWITHTHEFAMOUSTHEOREM,ITISLIKELYTHATTHEBABYLONIANSKNEWTHERESULTFORCERTAINSPECIFICTRIANGLESATLEASTAMILLENNIUMEARLIERTHANPYTHAGORASITISNOTKNOWNHOWTHEGREEKSORIGINALLYDEMONSTRATEDTHEPROOFOFTHEPYTHAGOREANTHEOREMIFTHEMETHODSOFBOOKIIOFEUCLIDSELEMENTSWEREUSED,ITISLIKELYTHATITWASADISSECTIONTYPEOFPROOFSIMILARTOTHEFOLLOWING“ALARGESQUAREOFSIDEABISDIVIDEDINTOTWOSMALLERSQUARESOFSIDESAANDBRESPECTIVELY,ANDTWOEQUALRECTANGLESWITHSIDESAANDBEACHOFTHESETWORECTANGLESCANBESPLITINTOTWOEQUALRIGHTTRIANGLESBYDRAWINGTHEDIAGONALCTHEFOURTRIANGLESCANBEARRANGEDWITHINANOTHERSQUAREOFSIDEABASSHOWNINTHEFIGURESTHEAREAOFTHESQUARECANBESHOWNINTWODIFFERENTWAYS1ASTHESUMOFTHEAREAOFTHETWORECTANGLESANDTHESQUARES2ASTHESUMOFTHEAREASOFASQUAREANDTHEFOURTRIANGLESNOW,SETTINGTHETWORIGHTHANDSIDEEXPRESSIONSINTHESEEQUATIONSEQUAL,GIVESTHEREFORE,THESQUAREONCISEQUALTOTHESUMOFTHESQUARESONAANDBBURTON1991THEREAREMANYOTHERPROOFSOFTHEPYTHAGOREANTHEOREMONECAMEFROMTHECONTEMPORARYCHINESECIVILIZATIONFOUNDINTHEOLDESTEXTANTCHINESETEXTCONTAININGFORMALMATHEMATICALTHEORIES,THEARITHMETICCLASSICOFTHEGNOMANANDTHECIRCULARPATHSOFHEAVENTHEPROOFOFTHEPYTHAGOREANTHEOREMTHATWASINSPIREDBYAFIGUREINTHISBOOKWASINCLUDEDINTHEBOOKVIJAGANITA,ROOTCALCULATIONS,BYTHEHINDUMATHEMATICIANBHASKARABHASKARASONLYEXPLANATIONOFHISPROOFWAS,SIMPLY,“BEHOLD“THESEPROOFSANDTHEGEOMETRICALDISCOVERYSURROUNDINGTHEPYTHAGOREANTHEOREMLEDTOONEOFTHEEARLIESTPROBLEMSINTHETHEORYOFNUMBERSKNOWNASTHEPYTHGOREANPROBLEMTHEPYTHAGOREANPROBLEMFINDALLRIGHTTRIANGLESWHOSESIDESAREOFINTEGRALLENGTH,THUSFINDINGALLSOLUTIONSINTHEPOSITIVEINTEGERSOFTHEPYTHAGOREANEQUATIONTHETHREEINTEGERSX,Y,ZTHATSATISFYTHISEQUATIONISCALLEDAPYTHAGOREANTRIPLESOMEPYTHAGOREANTRIPLESXYZ345512137242594041116061THEFORMULATHATWILLGENERATEALLPYTHAGOREANTRIPLESFIRSTAPPEAREDINBOOKXOFEUCLIDSELEMENTSWHERENANDMAREPOSITIVEINTEGERSOFOPPOSITEPARITYANDMNINHISBOOKARITHMETICA,DIOPHANTUSCONFIRMEDTHATHECOULDGETRIGHTTRIANGLESUSINGTHISFORMULAALTHOUGHHEARRIVEDATITUNDERADIFFERENTLINEOFREASONINGTHEPYTHAGOREANTHEOREMCANBEINTRODUCEDTOSTUDENTSDURINGTHEMIDDLESCHOOLYEARSTHISTHEOREMBECOMESINCREASINGLYIMPORTANTDURINGTHEHIGHSCHOOLYEARSITISNOTENOUGHTOMERELYSTATETHEALGEBRAICFORMULAFORTHEPYTHAGOREANTHEOREMSTUDENTSNEEDTOSEETHEGEOMETRICCONNECTIONSASWELLTHETEACHINGANDLEARNINGOFTHEPYTHAGOREANTHEOREMCANBEENRICHEDANDENHANCEDTHROUGHTHEUSEOFDOTPAPER,GEOBOARDS,PAPERFOLDING,ANDCOMPUTERTECHNOLOGY,ASWELLASMANYOTHERINSTRUCTIONALMATERIALSTHROUGHTHEUSEOFMANIPULATIVESANDOTHEREDUCATIONALRESOURCES,THEPYTHAGOREANTHEOREMCANMEANMUCHMORETOSTUDENTSTHANJUSTANDPLUGGINGNUMBERSINTOTHEFORMULATHEFOLLOWINGISAVARIETYOFPROOFSOFTHEPYTHAGOREANTHEOREMINCLUDINGONEBYEUCLIDTHESEPROOFS,ALONGWITHMANIPULATIVESANDTECHNOLOGY,CANGREATLYIMPROVESTUDENTSUNDERSTANDINGOFTHEPYTHAGOREANTHEOREMTHEFOLLOWINGISASUMMATIONOFTHEPROOFBYEUCLID,ONEOFTHEMOSTFAMOUSMATHEMATICIANSTHISPROOFCANBEFOUNDINBOOKIOFEUCLIDSELEMENTSPROPOSITIONINRIGHTANGLEDTRIANGLESTHESQUAREONTHEHYPOTENUSEISEQUALTOTHESUMOFTHESQUARESONTHELEGSFIGURE2EUCLIDBEGANWITHTHEPYTHAGOREANCONFIGURATIONSHOWNABOVEINFIGURE2THEN,HECONSTRUCTEDAPERPENDICULARLINEFROMCTOTHESEGMENTDJONTHESQUAREONTHEHYPOTENUSETHEPOINTSHANDGARETHEINTERSECTIONSOFTHISPERPENDICULARWITHTHESIDESOFTHESQUAREONTHEHYPOTENUSEITLIESALONGTHEALTITUDETOTHERIGHTTRIANGLEABCSEEFIGURE3FIGURE3NEXT,EUCLIDSHOWEDTHATTHEAREAOFRECTANGLEHBDGISEQUALTOTHEAREAOFSQUAREONBCANDTHATTHEAREOFTHERECTANGLEHAJGISEQUALTOTHEAREAOFTHESQUAREONACHEPROVEDTHESEEQUALITIESUSINGTHECONCEPTOFSIMILARITYTRIANGLESABC,AHC,ANDCHBARESIMILARTHEAREAOFRECTANGLEHAJGISHAAJANDSINCEAJAB,THEAREAISALSOHAABTHESIMILARITYOFTRIANGLESABCANDAHCMEANSANDTHEREFOREOR,ASTOBEPROVED,THEAREAOFTHERECTANGLEHAJGISTHESAMEASTHEAREAOFTHESQUAREONSIDEACINTHESAMEWAY,TRIANGLESABCANDCHGARESIMILARSOANDSINCETHESUMOFTHEAREASOFTHETWORECTANGLESISTHEAREAOFTHESQUAREONTHEHYPOTENUSE,THISCOMPLETESTHEPROOFEUCLIDWASANXIOUSTOPLACETHISRESULTINHISWORKASSOONASPOSSIBLEHOWEVER,SINCEHISWORKONSIMILARITYWASNOTTOBEUNTILBOOKSVANDVI,ITWASNECESSARYFORHIMTOCOMEUPWITHANOTHERWAYTOPROVETHEPYTHAGOREANTHEOREMTHUS,HEUSEDTHERESULTTHATPARALLELOGRAMSAREDOUBLETHETRIANGLESWITHTHESAMEBASEANDBETWEENTHESAMEPARALLELSDRAWCJANDBETHEAREAOFTHERECTANGLEAHGJISDOUBLETHEAREAOFTRIANGLEJAC,ANDTHEAREAOFSQUAREACLEISDOUBLETRIANGLEBAETHETWOTRIANGLESARECONGRUENTBYSASTHESAMERESULTFOLLOWSINASIMILARMANNERFORTHEOTHERRECTANGLEANDSQUAREKATZ,1993CLICKHEREFORAGSPANIMATIONTOILLUSTRATETHISPROOFTHENEXTTHREEPROOFSAREMOREEASILYSEENPROOFSOFTHEPYTHAGOREANTHEOREMANDWOULDBEIDEALFORHIGHSCHOOLMATHEMATICSSTUDENTSINFACT,THESEAREPROOFSTHATSTUDENTSCOULDBEABLETOCONSTRUCTTHEMSELVESATSOMEPOINTTHEFIRSTPROOFBEGINSWITHARECTANGLEDIVIDEDUPINTOTHREETRIANGLES,EACHOFWHICHCONTAINSARIGHTANGLETHISPROOFCANBESEENTHROUGHTHEUSEOFCOMPUTERTECHNOLOGY,ORWITHSOMETHINGASSIMPLEASA3X5INDEXCARDCUTUPINTORIGHTTRIANGLESFIGURE4FIGURE5ITCANBESEENTHATTRIANGLES2INGREENAND1INRED,WILLCOMPLETELYOVERLAPTRIANGLE3INBLUENOW,WECANGIVEAPROOFOFTHEPYTHAGOREANTHEOREMUSINGTHESESAMETRIANGLESPROOFICOMPARETRIANGLES1AND3FIGURE6ANGLESEANDD,RESPECTIVELY,ARETHERIGHTANGLESINTHESETRIANGLESBYCOMPARINGTHEIRSIMILARITIES,WEHAVEANDFROMFIGURE6,BCADSO,BYCROSSMULTIPLICATION,WEGETIICOMPARETRIANGLES2AND3FIGURE7BYCOMPARINGTHESIMILARITIESOFTRIANGLES2AND3WEGETFROMFIGURE4,ABCDBYSUBSTITUTION,CROSSMULTIPLICATIONGIVESFINALLY,BYADDINGEQUATIONS1AND2,WEGETFROMTRIANGLE3,ACAEECSOFIGURE8WEHAVEPROVEDTHEPYTHAGOREANTHEOREMTHENEXTPROOFISANOTHERPROOFOFTHEPYTHAGOREANTHEOREMTHATBEGINSWITHARECTANGLEITBEGINSBYCONSTRUCTINGRECTANGLECADEWITHBADANEXT,WECONSTRUCTTHEANGLEBISECTOROFBADANDLETITINTERSECTEDATPOINTFTHUS,BAFISCONGRUENTTODAF,AFAF,ANDBADASO,BYSAS,TRIANGLEBAFTRIANGLEDAFSINCEADFISARIGHTANGLE,ABFISALSOARIGHTANGLEFIGURE9NEXT,SINCEMEBFMABCMABF180DEGREESANDMABF90DEGREES,EBFANDABCARECOMPLEMENTARYTHUS,MEBFMABC90DEGREESWEALSOKNOWTHATMBACMABCMACB180DEGREESSINCEMACB90DEGREES,MBACMABC90DEGREESTHEREFORE,MEBFMABCMBACMABCANDMBACMEBFBYTHEAASIMILARITYTHEOREM,TRIANGLEEBFISSIMILARTOTRIANGLECABNOW,LETKBETHESIMILARITYRATIOBETWEENTRIANGLESEBFANDCABFIGURE10THUS,TRIANGLEEBFHASSIDESWITHLENGTHSKA,KB,ANDKCSINCEFBFD,FDKCALSO,SINCETHEOPPOSITESIDESOFARECTANGLEARECONGRUENT,BKAKCANDCAKBBYSOLVINGFORK,WEHAVEANDBKACCAKB（）THUS,BYCROSSMULTIPLICATION,THEREFORE,ANDWEHAVECOMPLETEDTHEPROOFTHENEXTPROOFOFTHEPYTHAGOREANTHEOREMTHATWILLBEPRESENTEDISONETHATBEGINSWITHARIGHTTRIANGLEINTHENEXTFIGURE,TRIANGLEABCISARIGHTTRIANGLEITSRIGHTANGLEISANGLECFIGURE11NEXT,DRAWCDPERPENDICULARTOABASSHOWNINTHENEXTFIGUREFIGURE12TRIANGLE1COMPARETRIANGLES1AND3TRIANGLE1GREENISTHERIGHTTRIANGLETHATWEBEGANWITHPRIORTOCONSTRUCTINGCDTRIANGLE3REDISONEOFTHETWOTRIANGLESFORMEDBYTHECONSTRUCTIONOFCDFIGURE13TRIANGLE1TRIANGLE3BYCOMPARINGTHESETWOTRIANGLES,WECANSEETHATCOMPARETRIANGLES1AND2TRIANGLE1GREENISTHESAMEASABOVETRIANGLE2BLUEISTHEOTHERTRIANGLEFORMEDBYCONSTRUCTINGCDITSRIGHTANGLEISANGLEDFIGURE14TRIANGLE1TRIANGLE2BYCOMPARINGTHESETWOTRIANGLES,WESEETHATBYADDINGEQUATIONS3AND4WEGETFROMFIGURES11AND12,WITHCD,WEHAVETHATPQCBYSUBSTITUTION,WEGETTHENEXTPROOFOFTHEPYTHAGOREANTHEOREMTHATWILLBEPRESENTEDISONEINWHICHATRAPEZOIDWILLBEUSEDFIGURE15BYTHECONSTRUCTIONTHATWASUSEDTOFORMTHISTRAPEZOID,ALL6OFTHETRIANGLESCONTAINEDINTHISTRAPEZOIDARERIGHTTRIANGLESTHUS,AREAOFTRAPEZOIDTHESUMOFTHEAREASOFTHE6TRIANGLESANDBYUSINGTHERESPECTIVEFORMULASFORAREA,WEGETWEHAVECOMPLETEDTHEPROOFOFTHEPYTHAGOREANTHEOREMUSINGTHETRAPEZOIDTHENEXTPROOFOFTHEPYTHAGOREANTHEOREMTHATIWILLPRESENTISONETHATCANBETAUGHTANDPROVEDUSINGPUZZLESTHESEPUZZLESCANBECONSTRUCTEDUSINGTHEPYTHAGOREANCONFIGURATIONANDTHEN,DISSECTINGITINTODIFFERENTSHAPESBEFORETHEPROOFISPRESENTED,ITISIMPORTANTTHATTHENEXTFIGUREISEXPLOREDSINCEITDIRECTLYRELATESTOTHEPROOFFIGURE16INTHISPYTHAGOREANCONFIGURATION,THESQUAREONTHEHYPOTENUSEHASBEENDIVIDEDINTO4RIGHTTRIANGLESAND1SQUARE,MNPQ,INTHECENTERSINCEMNANAMABEACHSIDEOFSQUAREMNPQHASLENGTHOFABTHISGIVESTHEFOLLOWINGAREAOFSQUAREONTHEHYPOTENUSESUMOFTHEAREASOFTHE4TRIANGLESANDTHEAREAOFSQUAREMNPQASMENTIONEDABOVE,THISPROOFOFTHEPYTHAGOREANTHEOREMCANBEFURTHEREXPLOREDANDPROVEDUSINGPUZZLESTHATAREMADEFROMTHEPYTHAGOREANCONFIGURATIONSTUDENTSCANMAKETHESEPUZZLESANDTHENUSETHEPIECESFROMSQUARESONTHELEGSOFTHERIGHTTRIANGLETOCOVERTHESQUAREONTHEHYPOTENUSETHISCANBEAGREATCONNECTIONBECAUSEITISA“HANDSON“ACTIVITYSTUDENTSCANTHENUSETHEPUZZLETOPROVETHEPYTHAGOREANTHEOREMONTHEIROWNFIGURE17TOCREATETHISPUZZLE,COPYTHESQUAREONBCTWICE,ONCEPLACEDBELOWTHESQUAREONACANDONCETOTHERIGHTOFTHESQUAREONACASSHOWNINFIGURE17PROOFUSINGFIGURE17TRIANGLECDEISCONGRUENTTOTRIANGLEACBBYLEGLEGINTRIANGLEACB,MACB90ANDTHESIDESHAVELENGTHSA,B,CINTRIANGLECDE,MCDE90ANDTHESIDESHAVELENGTHSA,B,CTRIANGLEEGHISCONGRUENTTOTRIANGLEACBBYLEGLEGTHEMEGH90ANDITSSIDESHAVELENGTHSAANDCSINCEEFBAAI,EGBTHUSTHEDIAGONALSCEANDEHAREBOTHEQUALTOCNOTEPIECES4AND7,ANDPIECES5AND6ARENOTSEPARATEDBYCALCULATINGTHEAREAOFEACHPIECE,ITCANBESHOWNTHATAREA1AREA2AREA3ARE